Algebra Examples

$f (x) = - 2 x^{3} - 8 x^{2} + 10 x$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - 2 x^{3} - 8 x^{2} + 10 x$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- 2 x^{3} - 8 x^{2} + 10 x = 0$ .

$- 2 x^{3} - 8 x^{2} + 10 x = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 2 x$ out of $- 2 x^{3} - 8 x^{2} + 10 x$ .

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Step 1.2.2.1.1

Factor $- 2 x$ out of $- 2 x^{3}$ .

$- 2 x \cdot x^{2} - 8 x^{2} + 10 x = 0$

Step 1.2.2.1.2

Factor $- 2 x$ out of $- 8 x^{2}$ .

$- 2 x \cdot x^{2} - 2 x (4 x) + 10 x = 0$

Step 1.2.2.1.3

Factor $- 2 x$ out of $10 x$ .

$- 2 x \cdot x^{2} - 2 x (4 x) - 2 x \cdot - 5 = 0$

Step 1.2.2.1.4

Factor $- 2 x$ out of $- 2 x (x^{2}) - 2 x (4 x)$ .

$- 2 x (x^{2} + 4 x) - 2 x \cdot - 5 = 0$

Step 1.2.2.1.5

Factor $- 2 x$ out of $- 2 x (x^{2} + 4 x) - 2 x (- 5)$ .

$- 2 x (x^{2} + 4 x - 5) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 1.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 1.2.2.2.1.2

Write the factored form using these integers.

$- 2 x ((x - 1) (x + 5)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$- 2 x (x - 1) (x + 5) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 1 = 0$

$x + 5 = 0$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.5.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.6

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 1.2.6.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.7

The final solution is all the values that make $- 2 x (x - 1) (x + 5) = 0$ true.

$x = 0, 1, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (1, 0), (- 5, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - 2 {(0)}^{3} - 8 {(0)}^{2} + 10 (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 2 \cdot 0^{3} - 8 {(0)}^{2} + 10 (0)$

Step 2.2.2

Remove parentheses.

$y = - 2 \cdot 0^{3} - 8 \cdot 0^{2} + 10 (0)$

Step 2.2.3

Remove parentheses.

$y = - 2 {(0)}^{3} - 8 {(0)}^{2} + 10 (0)$

Step 2.2.4

Simplify $- 2 {(0)}^{3} - 8 {(0)}^{2} + 10 (0)$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = - 2 \cdot 0 - 8 {(0)}^{2} + 10 (0)$

Step 2.2.4.1.2

Multiply $- 2$ by $0$ .

$y = 0 - 8 {(0)}^{2} + 10 (0)$

Step 2.2.4.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 - 8 \cdot 0 + 10 (0)$

Step 2.2.4.1.4

Multiply $- 8$ by $0$ .

$y = 0 + 0 + 10 (0)$

Step 2.2.4.1.5

Multiply $10$ by $0$ .

$y = 0 + 0 + 0$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (1, 0), (- 5, 0)$

y-intercept(s): $(0, 0)$

Step 4